Chaotic Dynamics and Monte Carlo Modelling
In practice, i.e., as long as the initial conditions cannot be specified exactlty, the outcome of a chaotic dynamical system can only be specified in statistical terms. Evolution equations (e.g., the Fokker-Planck equation) for a distribution of test particles can then be formulated, and as an alternative to analytical, mostly approximate or idealized solutions one may simulate the problem using Monte Carlo techniques. Such simulations are a well-known tool in the study of completely chaotic many-body systems such as star clusters or planetary rings, where the sample of test particles can indeed be taken to represent a zandom set of true solutions according to Bowen’s shadowing lemma. In this sense the Monte Carlo modelling plays a role analogous to that of averaging or mapping in regular dynamics, i.e.: the exact dynamical system is replaced by a model overlooking the details of the short-term motion but yielding a good approximation to the long-term behaviour. By a further discretization of the problem the stochastic system can be modelled as a Markov chain. Both Monte Carlo simulations and Markov models have been used in cometary dynamics, and we review some examples from this work to illustrate the success as well as limitations of these stochastic modelling techniques. Lyapunov characteristic exponents and Kolmogorov entropy appear to be suitable tools for estimating the underlying stochasticity to which Monte Carlo simulations refer.
Chaotic dynamics plays an important role for many problems of stellar dynamics (the classical N-body problem) and celestial mechanics (the dynamics of planetary ring particles and comets). Actually since the initial conditions cannot be specified exactly and also since the precision of integration is finite, the chaoticity of an orbit will cause exponential divergence of the computed orbit from the “true” orbit. Hence the outcome of a chaotic dynamical system can only be specified in statistical terms, and therefore a Monte Carlo approach becomes relevant. Indeed for the study of the dynamical evolution of globular clusters such an approach was undertaken already in the 1960’s (see Hénon 1973).
KeywordsMonte Carlo Simulation Globular Cluster Semimajor Axis Monte Carlo Modelling Oort Cloud
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